(a) Restricting both sides of \(\operatorname{Curl}V=E\)  to \(\Omega -O\)  yields \(\operatorname{Curl}V|_{\Omega -O}=0\) . Let \(V_3=E+(E^{(0,0)}/2\pi r)e_\theta \) , where \(E\in \mathcal {E}(\Omega ,\mathbb {R}^2)\) , such that \(\operatorname{Curl}V_3= E\)  [1]}. Hence \(\operatorname{Curl}(V_3-V)=0\)  which implies that there exists \(U \in \mathcal {B}(B_\epsilon )\) , with bulk density \(u\) , satisfying \(\nabla U = (V_3-V)|_{B_\epsilon }\) . Consequently \(\nabla u = (E^{(0,0)}/2\pi r)e_\theta -v_1\)  in \(\Omega -\lbrace S\cup \lbrace O\rbrace  \rbrace \)  and \(\llbracket u \rrbracket \nu =v_{2}\)  on \(\lbrace B_\epsilon \cap S\rbrace -\lbrace O\rbrace \) . The expression in () follows from integrating \(\nabla u\)  over the loop \(\partial B_\epsilon \) .
